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19-20 Show that the line integral is independent of path and evaluate the integral. 19. fc 2xe dx + (2y - x²e) dy, C is any path from (1, 0) to (2, 1)

19-20 Show that the line integral is independent of path and evaluate the integral. 19. fc 2xe dx + (2y - x²e) dy, C is any path from (1, 0) to (2, 1)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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& C: x= √t, y=t+1, z=1², 0≤t≤1 16. F(x, y, z) = (y²z + 2xz²)i + 2xyzj + (xy2 + 2x²z) k, 17. F(x, y, z) = yzei + e* j + xyek, C: r(t) = (t² + 1)i + (t² − 1)j + (t²- 2t) k, 0≤t≤2 18. F(x, y, z) = sin y i + (x cos y + cos z) j- y sin z k, C: r(t) sin ti + tj + 2tk, 0≤ t ≤TT/2 = evaluate the integral. 19-20 Show that the line integral is independent of path and 19. 2xe dx + (2y - x²e) dy, C is any path from (1, 0) to (2, 1) 20. fc sin y dx + (x cos y sin y) dy, C is any path from (2, 0) to (1, π) SECTION 16.3 The Fundament - 28. Let F = Vf, wher and C₂ that are no (a) Sc F. dr = C₁ 29. Show that if the conservative an derivatives, the 21. Suppose you're asked to determine the curve that requires the least work for a force field F to move a particle from one point to another point. You decide to check first whether F is conservative, and indeed it turns out that it is. How would you reply to the request? 22. Suppose an experiment determines that the amount of work required for a force field F to move a particle from the point (1, 2) to the point (5, -3) along a curve C₁ is 1.2 J and the work done by F in moving the particle along another curve C₂ between the same two points is 1.4 J. What can you say about F? Why? 23-24 Find the work done by the force field F in moving an object from P to Q. 23. F(x, y) = x³i+y³ j; P(1, 0), Q(2, 2) 24. F(x, y) = (2x + y)i + xj; P(1, 1), Q(4, 3) ap aç ду a 30. Use Exercise Scy dx + x conservative? = 31-34 Determi (b) connected, 31. {(x, y) | 33. {(x, y) | 34. {(x, y) | 35. Let FO (a) S (b) 36. (8
Transcribed Image Text:& C: x= √t, y=t+1, z=1², 0≤t≤1 16. F(x, y, z) = (y²z + 2xz²)i + 2xyzj + (xy2 + 2x²z) k, 17. F(x, y, z) = yzei + e* j + xyek, C: r(t) = (t² + 1)i + (t² − 1)j + (t²- 2t) k, 0≤t≤2 18. F(x, y, z) = sin y i + (x cos y + cos z) j- y sin z k, C: r(t) sin ti + tj + 2tk, 0≤ t ≤TT/2 = evaluate the integral. 19-20 Show that the line integral is independent of path and 19. 2xe dx + (2y - x²e) dy, C is any path from (1, 0) to (2, 1) 20. fc sin y dx + (x cos y sin y) dy, C is any path from (2, 0) to (1, π) SECTION 16.3 The Fundament - 28. Let F = Vf, wher and C₂ that are no (a) Sc F. dr = C₁ 29. Show that if the conservative an derivatives, the 21. Suppose you're asked to determine the curve that requires the least work for a force field F to move a particle from one point to another point. You decide to check first whether F is conservative, and indeed it turns out that it is. How would you reply to the request? 22. Suppose an experiment determines that the amount of work required for a force field F to move a particle from the point (1, 2) to the point (5, -3) along a curve C₁ is 1.2 J and the work done by F in moving the particle along another curve C₂ between the same two points is 1.4 J. What can you say about F? Why? 23-24 Find the work done by the force field F in moving an object from P to Q. 23. F(x, y) = x³i+y³ j; P(1, 0), Q(2, 2) 24. F(x, y) = (2x + y)i + xj; P(1, 1), Q(4, 3) ap aç ду a 30. Use Exercise Scy dx + x conservative? = 31-34 Determi (b) connected, 31. {(x, y) | 33. {(x, y) | 34. {(x, y) | 35. Let FO (a) S (b) 36. (8
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